On Generators and Representations of the Sporadic Simple Groups

Abstract

In this paper we determine the irreducible projective representations of sporadic simple groups over an arbitrary algebraically closed field F , whose image contains an almost cyclic matrix of prime-power order. A matrix M is called cyclic if its characteristic and minimum polynomials coincide, and we call M almost cyclic if, for a suitable α ∈ F , M is similar to diag(α · Id h , M1), where M1 is cyclic and 0 ≤ h ≤ n. The paper also contains results on the generation of sporadic simple groups by minimal sets of conjugate elements.

Almost cyclic elements in cross-characteristic representations of finite groups of Lie type

“…For n > 2, we get that φ(g) is not almost cyclic by Lemma 2.3 once we show that 2n|g| < q n −1 2 . Now, 2n|g| ≤ 2n · 2 t+1 (q + 1) ≤ 4n 2 (q + 1) and 4n 2 (q + 1) < q n −1 2 for all n ≥ 3 and q odd, unless (n, q) ∈ {(3, 3), (3,5), (3,7), (4, 3), (4, 5), (5, 3), (6, 3)}. For L as listed in the table below, using the GAP package we get the following: Thus in these cases, by Lemma 2.3, we conclude that φ(g) is not almost cyclic.…”

“…where p m = |Φ| p . Then, by Lemma 3.1, |g| ≤ p t+m |q (since e < p).If m ≥ 1 and (n, q) = (4, 8),(5,8), then n 2 p m (q…”

“…In both cases, φ(g) cannot be almost cyclic. (2) L = Ω(7,5). Here |g| ≤ 31, α(L) ≤ 7 and dim φ ≥ 649, whence 649 > 7(31 − 1) = 210.…”

“…By the above, we are left to examine the following cases: Proof. Suppose that φ(g) is almost cyclic and assume that (n, q) = (4, 2), (4, 4), (5, 2), (5,3). Since dim φ ≥ (q n +1)(q n−1 −q) q 2 −1 − 1, the following bound must be met:…”

“…If L = P Ω − (10, 2), then |g| ∈ {3, 4,5,7,8,9,11,16, 17} and dim φ ≥ 151. If |g| ≤ 16, then α(g) ≤ 10, whence 151 > 10(16 − 1) = 150, a contradiction.…”